Lecture 34.

We discussed canonical quantization of Fermi fields (that is, fields
obeying a Dirac equation). To obtain positive definite energies after
quantization, we imposed canonical *anticommutation* relations, which
also led to the Pauli exclusion principle.

We first displayed the complete set of plane wave solutions to the
Dirac equation, which involved the fixed spinor column vectors
u_p^(1), u_p^(2), v_p^(1), and v_p^(2). We used these to write the
most general classical solution for the bispinor field \psi as a
Fourier integral over all plane wave solutions. As for the complex
Klein-Gordon field, we allow \psi to be complex, which means that
complex conjugate plane wave solutions need not have exactly conjugate
coefficients.  There are thus 2 coefficients b_k^(r) --- 1 for each
positive frequency solution u_k^(r) --- and 2 independent coefficients
c_k^(r)* --- 1 for each negative frequency solution v_k^(r). (Here the
* is chosen by convention, whenever the solution has negative
frequency).

Upon quantization, the coefficients all become creation and
annihilation operators.  The extra set of allowed coefficients become,
on quantization, creation operators for a whole new set of particles.
As in the complex Klein-Gordon case, these new particles can be shown
to have the same mass, but opposite Noether charge (associated with
the field's phase symmetry), to the original particles. Thus we
obtain, upon quantization, creation and annihilation operators for
particles and antiparticles in the spin states (r).

We calculated the Hamiltonian, finding that --- unlike the
Klein-Gordon case --- adopting the field-dependent Hamiltonian does
not automatically eliminate the problem of states with negative
contributions to the energy. Instead, we saw that negative frequency
solutions make negative contributions to the classically-computed
energy. Upon quantization, this problem persisted --- while the
particles associated with the b creation and annihilation operators
each contribute the positive energy \hbar \omega_k, those associated
with the c creation and annihilation operators naively seem to each
contribute the negative energy -\hbar \omega_k.

To get rid of this undesirable sign, we postulated that fermions obey
canonical *anticommutation* relations. This made the number
operators for the c-type particles (that is, the antiparticles)
appear in the Hamiltonian with positive sign. Again, to get rid of
infinite zero point energies, we have to normal order --- now defined
as reordering to get all annihilation operators on the right, then
discarding all the *anticommutators* that were generated in the
reordering.

We showed that, even though they no longer have harmonic oscillator
commutation relations with each other, the creation and annihilation
operators still have harmonic oscillator commutation relations with
the Hamiltonian --- which means that they still raise and lower the
energy by the quantum \hbar \omega_k. So the Fock space --- again
spanned by energy eigenstates --- again contains a vacuum with no
particles, which is annihilated by all the annihilation operators, as
well as states with particles and antiparticles in them, obtained
mathematically by operating on the vacuum with various creation
operators.

However, a major difference from the Fock space of bosons arises when
we try to apply the creation operator for a single state twice, to
doubly occupy a single particle or antiparticle state. Since the
creation operator anticommutes with itself, its square must be zero:
that is, applying the creation operator twice to the vacuum always
annihilates it. Thus each state k, (r) leads to a Hilbert space that,
unlike the harmonic oscillator, only has two states: an empty state,
which can be raised once to give a 1-particle state, but which is then
the highest state, annihilated when we try to raise it again.

This fact, that only 1 particle can be put in a single state, is just
the Pauli exclusion principle. The Dirac equation, along with 
canonical anticommutation relations, has automatically led to a theory
with multiple particles and antiparticles, which are all fermions. The
anticommutation relations also lead to the required antisymmetry of
fermion wavefunctions.

Finally, we discussed the classical limit of quantum fields. We noted
that Bose fields have both a particle limit, with finite number of
arbitrarily high frequency particles; and a field limit, where large
numbers of finite frequency particles have wave functions that add
coherently. Fermions only have the particle limit, since the occupancy
of any state is bounded by 1. Fermion fields also anticommute, even in
the classical limit.


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KB